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Journal Article•DOI•

Simultaneous development of velocity and temperature profiles for laminar flow of a non-Newtonian fluid in the entrance region of flat ducts

Joseph Yau1, Chi Tien1•
University of Windsor1
01 Aug 1963-Canadian Journal of Chemical Engineering (Wiley Subscription Services, Inc., A Wiley Company)-Vol. 41, Iss: 4, pp 139-145
TL;DR: In this paper, the authors used the momentum and energy integral method of von Karman and Pohlhausen to solve the entrance heat transfer problem for a non-Newtonian fluid in a flat duct.
Abstract: The momentum and energy integral method of von Karman and Pohlhausen is used to solve the entrance heat transfer problem for a non-Newtonian fluid in a flat duct. The initial temperature and velocity profiles are assumed to be flat. The fluid is assumed to obey the Ostwald-de Wael model and its physical properties are assumed to be constant. Dimensionless expressions for temperature and velocity profiles are obtained by numerical methods. The results of this investigation indicate that, similar to the case of Newtonian fluid, the parameters which influence entrance heat transfer are x/b ratio, Reynolds number and Prandtl number, provided these groups are properly defined. La methode tenant compte du momentum et de l'energie de von Karman et de Pohlhausen est utilisee pour resoudre le probleme de la transmission de chaleur a l'entree d'un conduit plat pour un fluide non-Newtonien. Les profils initiaux de temperature et vitesse sont supposes uniformes. Le fluide est conforme au modele Ostwald-de Wael et ses proprieties sont supposees constantes. Les expressions sans dimensions sont obtenues a l'aide d'analyses numeriques pour les profils de temperature et vitesse. Les resultats de cette recherche indiquent que, similarement au cas des fluides non-Newtoniens, les parametres influencant la transmission de chaleur a l'entree, sont x/b, Re et Pr en autant que ces groupes son definis adequatement.

Summary (1 min read)

Jump to: [B. Assumptions] – [oCU* (3>1*Z) ( n t] – [Definition of the Nusselt Number] – [B. Entrance length] – [The values in] and [B U M]

B. Assumptions

  • The assumptions used in this thesis are: 1. The flow is two dimensional.
  • The fluid is incompressible and has constant physical properties.

oCU* (3>1*Z) ( n t

  • Conventionally, the total pressure drop between the inlet and any point in the fully developed region Eq. (3-2) and (4-2) can be substituted into (4-1) which in turn will lead to an expression for A.
  • The inte gral on the left hand side of Eq. (4-1) would yield different expressions depending on the relative magnitude of Ô and A .

Definition of the Nusselt Number

  • If the following dimensionless quantities are introduced, In this thesis the velocity distribution given by Eq. (5-2) coincides with the fully developed profile when 5 = b, but its approach is not asymptotic.
  • It is believed that the boundary layer approach is not suitable as velo city becomes fully developed.
  • Collins and Schowalter (5) used another method to obtain more exact expressions.
  • No apparent difference between their results and those obtained in this work is detected when compared.

B. Entrance length

  • This means that the velocity approaches its fully developed value finitely.
  • But the actual approach should be asymptotic.
  • Some authors define entrance length dif ferently.
  • They choose the distance required for the centerline velocity to reach an arbitrary percentage (usually 99^) of its fully developed value as the entrance length.
  • This gives an entrance length larger than that obtained by the "finite" definition.

The values in

  • Step 1 are the starting values which were found previously.
  • The increment h should be small but its value may be increased later to speed up the integration.

B U M

  • Reproduced with permission of the copyright owner.
  • Reproduced with permission of the copyright owner.

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SIMULTANEOUS DEVELOPMENT OP VELOCITY AND TEMPERATURE PROFILES
FOR LAMINAR FLOW OF A NON-NEWTONIAN FLUID
IN THE ENTRANCE REGION OF FLAT DUCTS
A Thesis
Submitted to the Faculty of graduate Studies through
the Department of Chemical Engineering in Partial
Fulfillment of the Requirements for the Degree
of Master of Applied Science at Assumption
University of Windsor
by
JOSEPH Y. YAU
B.A.Sc., Assumption University of Windsor, 1961
Windsor, Ontario, Canada
1962
ASSUMPTION UNIVERSITY LIBRflRV
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
UMI Number: EC52501
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ABSTRACT
A theoretical analysis for the laminar non-Newtonian
fluid flow in the entrance region of a flat duct is presented
in this thesis. The non-Newtonian fluid is assumed to be of
the Ostwald-de Waele model and its physical properties are
assumed to be constant. The initial velocity and temperature
profiles of the fluid prior to its entry are considered to
be flat; and the walls of the duct are maintained at uniform
but different temperatures. The momentum and energy integral
method of von Karman and Pohlhausen is applied for the solu
tion of entrance heat transfer problems. Dimensionless
expressions for velocity and temperature profiles, as well
as pressure loss and Nusselt's modulus are obtained from
numerical methods.
The results of this thesis indicate that, as in the
case of Newtonian fluid, the parameters which influence
entrance heat transfer are L/D ratio, Reynolds number and
Prandtl number, provided these groups are properly defined
for non-Newtonian fluids.
iii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Citations
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Book Chapter•DOI•
Heat transfer to newtonian and non-newtonian fluids in rectangular ducts

[...]

J.P. Hartnett1, Milivoje M. Kostic1•
University of Illinois at Chicago1
01 Jan 1989-Advances in heat transfer
TL;DR: In this paper, the authors provide an overview of the analytical and experimental hydrodynamics and heat transfer studies of Newtonian and non-Newtonian fluids in laminar and turbulent flow through rectangular tubes.
Abstract: Publisher Summary This chapter provides an overview of the analytical and experimental hydrodynamics and heat transfer studies of Newtonian and non-Newtonian fluids in laminar and turbulent flow through rectangular tubes. The chapter in particular focuses on the rectangular duct geometry, with emphasis on the friction factor and heat transfer behavior of non-Newtonian fluids. It is recognized that non-Newtonian behavior is generally more complicated than Newtonian flow. In the case of non-Newtonian fluids, the theoretical predictions yield low estimates of the heat transfer under laminar flow conditions. The fact that the available experimental heat transfer measurements lie above the predictions could reflect an inadequacy in the analytical model. For non-Newtonians in turbulent flow through rectangular channels, the situation is even more complicated. Some non-Newtonian fluids act as pseudoplastics, showing some reduction in friction and heat transfer as compared with a Newtonian fluid. Other non-Newtonian fluids experience large reductions in the friction factor and in heat transfer under turbulent flow conditions.

262 citations

Book Chapter•DOI•
Heat Transfer in Non-Newtonian Fluids

[...]

A.B. Metzner1•
University of Delaware1
01 Jan 1965-Advances in heat transfer
TL;DR: This chapter provides the general survey of rheological behavior necessary for an understanding of transport processes in non-Newtonian fluids.
Abstract: Publisher Summary This chapter focuses on heat transfer in non-Newtonian fluids. This chapter illustrates that useful engineering studies which deal with transport processes in non-Newtonian fluids have been underway for the past several years. Earlier heat transfer reviews therefore serves primarily to separate a very small number of useful contributions from the substantial mass of trivial, misleading or even incorrect concepts and interpretations available at that time. This chapter provides the general survey of rheological behavior necessary for an understanding of transport processes in non-Newtonian fluids. The fluid classification system of interest is based on the form of the constitutive equations used to describe the properties of real fluids quantitatively. The chapter discusses all or more recent publication of work on heat transfer to non-Newtonian fluids since 1954 and provides an overview of the current state of the art. The chapter also describes fluid mechanical studies dealing with non-Newtonian materials.

99 citations

Journal Article•DOI•
Laminar forced convection heat transfer of a non-newtonian fluid in a square duct

[...]

Ashok R. Chandrupatla1, V.M.K. Sastri1•
Indian Institute of Technology Madras1
01 Dec 1977-International Journal of Heat and Mass Transfer
TL;DR: In this article, numerical solutions for laminar heat transfer of a non-Newtonian fluid in the thermal entrance region of a square duct are presented for three thermal boundary conditions.

85 citations

Journal Article•DOI•
Advances in Chemical Engineering

[...]

J. F. Richardson
01 Jul 1965-Nature
TL;DR: Drew, Hoopes, Vermeulen, and Cokelet as discussed by the authors presented a survey of advances in Chemical Engineering Vol. 5, No. 5. Pp. x + 317.
Abstract: Advances in Chemical Engineering Vol. 5. Edited by Thomas B. Drew, John W. Hoopes jun., Theodore Vermeulen and Giles R. Cokelet. Pp. x + 317. (New York: Academic Press, Inc.; London: Academic Press, Inc. (London), Ltd., 1964.) 100s.

67 citations

Book Chapter•DOI•
Heat and mass transfer in rheologically complex systems

[...]

R.P. Chhabra1•
Indian Institute of Technology Kanpur1
01 Jan 1999
TL;DR: In this paper, the authors discuss the properties of non-Newtonian materials in terms of heat and mass transport in processes involving rheologically complex materials, especially with regard to the implications of shear dependent viscosity and viscoelasticity.
Abstract: Publisher Summary This chapter discusses the phenomena of heat and mass transport in processes involving rheologically complex materials, especially with regard to the implications of shear dependent viscosity and viscoelasticity of materials. The highly non-linear and strongly coupled nature of the field equations preclude the possibility of analytical results, and therefore, usually numerical solutions are sought, even for as simple a situation as that of laminar flow of a purely viscous fluid in a circular tube. The thermo-physical properties of the commonly used non-Newtonian materials are described. It is clear that in the laminar region the shear thinning enhances heat transfer and reduces pressure drop as compared with a Newtonian fluid. The temperature dependent viscosity facilitates the heating of fluids while it impedes the efficacy of cooling of fluids for shear thinning materials. While from a heat transfer standpoint, the circular geometry is the most efficient one but when pressure loss is also of concern, short pipe lengths are preferred.

30 citations

References
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Journal Article•DOI•
Momentum and heat transfer in laminar boundary-layer flows of non-Newtonian fluids past external surfaces

[...]

Andreas Acrivos1, M. J. Shah1, Eugene E. Petersen1•
University of California, Berkeley1
01 Jun 1960-Aiche Journal
TL;DR: In this article, a theoretical analysis for the laminar flow past arbitrary external surfaces of non-Newtonian fluids of the power-law model is presented, where the main problem which is considered is how to predict the drag and the rate of heat transfer from an isothermal surface to the fluid.
Abstract: A theoretical analysis for the laminar flow past arbitrary external surfaces of non-Newtonian fluids of the power-law model is presented. The main problem which is considered is how to predict the drag and the rate of heat transfer from an isothermal surface to the fluid. Inspectional analysis of the modified boundary-layer equations yields a general relationship both for the drag coefficient and for the Nusselt number as functions of the generalized Reynolds and Prandtl numbers. The flow past a horizontal flat plate is studied in detail numerically.

382 citations

Journal Article•DOI•
Steady Flow in the Transition Length of a Straight Tube

[...]

Henry L. Langhaar1•
Purdue University1
01 Jun 1942-Journal of Applied Mechanics

334 citations

Journal Article•DOI•
The application of boundary-layer theory to power-law pseudoplastic fluids: Similar solutions

[...]

W. R. Schowalter1•
Princeton University1
01 Mar 1960-Aiche Journal
TL;DR: In this paper, the types of potential flows necessary for similar solutions to the boundary-layer equations have been determined for two-dimensional and three-dimensional non-Newtonian fluids.
Abstract: Two- and three-dimensional boundary-layer equations have been developed for pseudoplastic non-Newtonian fluids which can be characterized by a power-law relationship between shear stress and velocity gradient. The types of potential flows necessary for similar solutions to the boundary-layer equations have been determined. For two-dimensional flow the results are similar to those obtained for Newtonian fluids. For three-dimensional flow, however, the possibility of similar solutions depends on the nature of the expression which describes effective viscosity of the fluid. At most, similar solutions are possible only for the case of flow past a flat plate where the potential velocity vector is not perpendicular to the leading edge of the plate; this is a much more restrictive condition than is obtained for Newtonian fluids.

280 citations

Journal Article•DOI•
Advances in Chemical Engineering

[...]

J. F. Richardson
01 Jul 1965-Nature
TL;DR: Drew, Hoopes, Vermeulen, and Cokelet as discussed by the authors presented a survey of advances in Chemical Engineering Vol. 5, No. 5. Pp. x + 317.
Abstract: Advances in Chemical Engineering Vol. 5. Edited by Thomas B. Drew, John W. Hoopes jun., Theodore Vermeulen and Giles R. Cokelet. Pp. x + 317. (New York: Academic Press, Inc.; London: Academic Press, Inc. (London), Ltd., 1964.) 100s.

67 citations

Journal Article•DOI•
Behavior of non-Newtonian fluids in the inlet region of a channel

[...]

Morton Collins1, W. R. Schowalter1•
Princeton University1
01 Jan 1963-Aiche Journal

37 citations

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